In June last year, the High Court ruled Ceredigion county council had acted unlawfully when it refused to pay for the youngsters' daily bus journeys to and from the Welsh-language school nearest their homes.
Ysgol Preseli is eight miles away, but across the border in Pembrokeshire.
Ceredigion is now battling to overturn the judge's decision, claiming it will have "absurd consequences" for local authorities.
The council is prepared to pay for free bus rides for Matthew Jones, Byron Rees and Aled James to attend Dyffryn Teifi, a Welsh-medium school in Ceredigion. But it is 18 miles away from the boys' homes, in Llandysul.
Pembrokeshire county council runs buses to their current school, Ysgol Preseli, but insists that, as Ceredigion residents, the boys must pay their way.
In June last year, Mr Justice Collins ruled Ceredigion was legally obliged to pay the bus bills under the 1996 Education Act. If it did not, the boys'
parents would in theory have a good defence to a criminal truancy charge if their children failed to attend school regularly (because free transport had not been provided to the nearest suitable school).
But Ceredigion's barrister, Mr Nigel Giffin QC, this week argued the judge's interpretation of the law was "simply wrong". He said it was obvious from the wording of the law that all the council had to do was make a rational decision as to whether providing free transport to Ysgol Preseli was "necessary" to ensure the boys received a "suitable" education. That, said Mr Giffin, was what Ceredigion had done.
The result of Mr Justice Collins's decision, added the barrister, was "nothing short of absurd" for local authorities as, in most cases, the possibility of parents being prosecuted for failing to ensure their children attended school regularly was entirely theoretical.
However, Mr Andrew Nicol QC, for the three boys, said Mr Justice Collins had been right to rule that it is always necessary for a council to lay on free school transport where, otherwise, parents would have a defence to a truancy charge.
The Appeal Court reserved its decision to an unspecified date.